# Quasitruncated cuboctahedron

Quasitruncated cuboctahedron
Rank3
TypeUniform
Notation
Bowers style acronymQuitco
Coxeter diagramx4/3x3x ()
Elements
Faces12 squares, 8 hexagons, 6 octagrams
Edges24+24+24
Vertices48
Vertex figureScalene triangle, edge lengths 2, 3, 2–2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13-6{\sqrt {2}}}}{2}}\approx 1.06239}$
Volume${\displaystyle 2(11-7{\sqrt {2}})\approx 2.20101}$
Dihedral angles8/3–4: 135°
8/3–6: ${\displaystyle \arccos \left({\frac {\sqrt {3}}{3}}\right)\approx 54.73561^{\circ }}$
6–4: ${\displaystyle \arccos \left({\frac {\sqrt {6}}{3}}\right)\approx 35.26439^{\circ }}$
Central density-1
Number of external pieces146
Level of complexity26
Related polytopes
ArmySemi-uniform Girco, edge lengths ${\displaystyle {\sqrt {2}}-1}$ (octagons), ${\displaystyle 2-{\sqrt {2}}}$ (ditrigon-rectangle)
RegimentQuitco
DualGreat disdyakis dodecahedron
ConjugateGreat rhombicuboctahedron
Convex coreOctahedron
Abstract & topological properties
Flag count288
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryB3, order 48
Flag orbits6
ConvexNo
NatureTame

The quasitruncated cuboctahedron or quitco, also called the great truncated cuboctahedron, is a uniform polyhedron. It consists of 12 squares, 8 hexagons, and 6 octagrams, with one of each type of face meeting per vertex. It can be obtained by quasicantitruncation of the cube or octahedron, or equivalently by quasitruncating the vertices of a cuboctahedron and then adjusting the edge lengths to be all equal.

## Vertex coordinates

A quasitruncated cuboctahedron of edge length 1 has vertex coordinates given by all permutations of:

• ${\displaystyle \left(\pm {\frac {2{\sqrt {2}}-1}{2}},\,\pm {\frac {{\sqrt {2}}-1}{2}},\,\pm {\frac {1}{2}}\right)}$.